multiple-criteria sorting using electre tri assistant

Multiple-criteria sorting

using ELECTRE TRI Assistant

Roman Słowiński

Poznań University of Technology, Poland

Roman Słowiński

2

Weak points of preference modelling using utility (value) function

Utility function distinguishes only 2 possible relations between actions:

preference relation: a b U(a) > U(b)

indifference relation: a b U(a) = U(b)

is asymmetric (antisymmetric and irreflexive) and transitive

is symmetric, reflexive and transitive

Transitivity of indifference is troublesome, e.g.

In consequence, a non-zero indifference threshold qi is necessary

An immediate transition from indifference to preference is unrealistic,

so a preference threshold pi qi and a weak preference relation

are desirable

Another realistic situation which is not modelled by U is incomparability,

so a good model should include also an incomparability relation ?

?

3

Four basic preference relations and an outranking relation S

Four basic preference relations are: {, , , ?}

Criterion with thresholds pi(a)qi(a)0 is called pseudo-criterion

The four basic situations of indifference, strict preference, weak

preference, and incomparability are sufficient for establishing

a realistic model of Decision Maker’s (DM’s) preferences

gi(b)gi(a)gi(a)-pi(b) gi(a)-qi(b) gi(a)+qi(a) gi(a)+pi(a)

ab ba baabab

preference

4

Axiom of limited comparability (Roy 1985):

Whatever the actions considered, the criteria used to compare them,

and the information available, one can develop a satisfactory model

of DM’s preferences by assigning one, or a grouping of two or three

of the four basic situations, to any pair of actions.

Outranking relation S groups three basic preference relations:

S = {, , } – reflexive and non-transitive

aSb means: „action a is at least as good as action b”

For each couple a,bA:

aSb non bSa ab ab

aSb bSa ab

non aSb non bSa a?b

Four basic preference relations and an outranking relation S

5

ELECTRE TRI: sorting problem (P)

Class 1

...

x x xx x

x x x xx

x x xx

x

x

x

x x

xx

xx x x x

x x

x x xx x x

A

Class 2

Class p

Class p Class p-1 ... Class 1

6

ELECTRE TRI

Input data: finite set of actions A={a, b, c, …, h}

consistent family of criteria G={g1, g2, …, gn}

preference-ordered decision classes Clt, t=1,…,p

Decision classes are caracterized by limit profiles bt, t=0,1,…,p

The preference model, i.e. outranking relation S is constructed for

each couple (a, bt), for every aA and t=0,1,…,p

g1

g2

g3

gn

b0 b1 bp-2 bp-1 bpb2

Cl1 Cl2 Clp-1 Clp

a d

7

ELECTRE TRI

Preferential information (i=1,…,n)

intracriteria: – indifference thresholds qit for each class limit, t=0,1,

…,p

– preference thresholds pit for each class limit, t=0,1,

…,p

intercriteria: – importance coefficients (weights) of criteria ki

– veto thresholds for each class limit vit, t=0,1,…,p

0qitpi

tvit are constant for each class limit bt, t=0,1,…,p

Concordance and discordance tests of ELECTRE TRI validate or

invalidate the assertions aSbt and btSa

8

ELECTRE TRI - concordance test

Checks how strong is the coalition of criteria concordant with

the hypothesis aSbt (for each couple (a,bt), aA and t=0,1,…,p)

Concordance coefficient Ci(a,bt) for each criterion gi:

gi(bt) gi(a)gi(bt)-pit gi(bt)-qi

t gi(bt)+qit gi(bt)+pi

t

preferenceCi(a,bt)

0

1

cost-type

gi(bt)

preferenceCi(a,bt)

0

1

gain-type

gi(bt)-pit gi(bt)-qi

t gi(bt)+qit gi(bt)+pi

t gi(a)

9

ELECTRE TRI - concordance test

Aggregation of concordance coefficients for a,bA:

C(a,bt) [0, 1]

ni i

ni tii

tk

b,aCkb,aC

1

1

10

ELECTRE TRI - discordance test

Checks how strong is the coalition of criteria discordant with

the hypothesis aSbt (for each couple (a,bt), aA and t=0,1,…,p)

Discordance coefficient Di(a,bt) for each criterion gi:

gi(a)0

1

Ci(a,bt)preference

cost-type

gi(bt)+vit

Di(a,bt)

gi(bt)-pit gi(bt)-qi

t gi(bt) gi(bt)+qit gi(bt)+pi

t

gi(a)

preferenceDi(a,bt)

0

1

gain-type

Ci(a,bt)

gi(bt)-vit gi(bt)gi(bt)-pi

t gi(bt)-qit gi(bt)+qi

t gi(bt)+pit

11

ELECTRE TRI – credibility of the outranking relation

Conclusion from concordance and discordance tests – credibility of

the outranking relation:

(a,bt)[0, 1]

What is the assignment of actions to decision classes ?

tti

Fi t

titt

b,aCb,aDiF

b,aCb,aD

b,aCb,a

: where

1

1

(a,bt)

(bt,a)(bt,a)

a btbt{}aa ? bt a{}bt

not

not not

yes

yes yes

12

ELECTRE TRI – exploitation of S and final recommendation

Assignment of actions to decision classes based on two ways of

comparison of actions aA to limit profiles bt, t=0,1,…,p

Pessimistic: compare action a successively to each profile bt,

t=p-1,…,1,0; if bt is the first profile such that aSbt, then aClt+1

Pessimistic, because for zero thresholds, aClt+1 gi(a)gi(bt) i,

e.g. aCl1

g1

g2

g3

gn

b0 b1 bp-2 bp-1 bpb2

Cl1 Cl2 Clp-1 Clp

a

comparison of action a to profiles bt

13

ELECTRE TRI – exploitation of S and final recommendation

Assignment of actions to decision classes based on two ways of

comparison of actions aA to limit profiles bt, t=0,1,…,p

Optimistic: compare action a successively to each profile bt,

t=1,…,p; if bt is the first profile such that bt{ }a, then aClt

Optimistic, because for zero thresholds, aClt gi(bt)>gi(a) i,

e.g. aCl2

g1

g2

g3

gn

b0 b1 bp-2 bp-1 bpb2

Cl1 Cl2 Clp-1 Clp

a

comparison of action a to profiles bt

14

ELECTRE TRI - example

kPRICE=3kTIME=5kCOMFORT=2

=0.75

15

ELECTRE TRI - example

??

16

ELECTRE TRI - example

=0.75

17

Inferring an ELECTRE TRI model from assignment examples

An ELECTRE TRI model M is composed of:

limit profiles bt, t=0,1,…,p

importance coefficients (weights) of criteria ki, i=1,…,n

indifference and preference thresholds qit, pi

t, i=1,…,n, t=0,1,…,p

veto thresholds vit, i=1,…,n, t=0,1,…,p

assignment procedure – either pessimistic or optimistic

We will apply a regression approach to infer an ELECTRE TRI model

compatible with a set of assignment examples

Inferring all elements of the model is computationnally hard and,

moreover, the more elements are unconstrained, the larger is the set

of all compatible models – then the choice of one model is uneasy

We assume:

no veto phenomenon occurs, i.e. vit=, i=1,…,n, t=0,1,…,p

pessimistic assignment procedure

18

Inferring an ELECTRE TRI model from assignment examples

General inference scheme of ELECTRE TRI Assistant:

AR

AR

19

Inferring an ELECTRE TRI model from assignment examples

Let us suppose aj DM Clt, Clt=[bt-1, bt]

aj DM Clt ajSbt-1 not ajSbt

i.e. (aj,bt-1) and (aj,bt)<

We define slack variables xaj0 and yaj>0, such that:

(aj,bt-1) xaj =

(aj,bt) + yaj =

If xaj0 and yaj>0, then

aj M Clt ’[yaj, +xaj]

We consider set AR={a1, a2,…, ar} of reference actions,

and assignment examples provided by the DM:

aj DM Clt , j=1,…,r

20

Inferring an ELECTRE TRI model from assignment examples

The regression problem will include the following variables:

xaj and yaj, j=1,…,r (slack variables – 2r)

(cutting level – 1)

ki, i=1,…,n (weights – n)

bit, i =1,…,n, t=0,1,…,p (limit profiles – np)

qit, i=1,…,n, t=0,1,…,p (indifference thresholds – np)

pit, i=1,…,n, t=0,1,…,p (preference thresholds – np)

The objective of the regression problem:

model M will be consistent with assignment examples iff

xaj0 and yaj>0, for all j=1,…,r

the objective should be to maximize ajaj

Aay,x

Rj min

21

Inferring an ELECTRE TRI model from assignment examples

If , then all reference actions are „correctly”

assigned by model M, for all

Objective takes into account the „worst case” only

To improve the second, third, … „worst case”, we consider objective

where is a small positive value Maximization of the new objective can be rewritten as:

aj

Aaaj

Aaxy'

Rj

Rj

min , min

R

jR

j Aaajajajaj

Aayxy,xmin

Rjaj

Rjaj

Aaajaj

Aay

Aax

y,xR

j

,

, s.t.

Max

ajajAa

y,xR

j min

0min

ajajAa

y,xR

j

>

22

Inferring an ELECTRE TRI model from assignment examples

The regression problem:

r

r

r

r

2

np

n(p+1)

n+n(p+1)

number of constraints& variables:

p,...,,tk,...,iqk

p,...,,tk,...,iqp

p,...,,tk,...,ippbgbg

,.

Aayb,a

Aaxb,a

Aay

Aax

y,x

tii

ti

ti

ti

tititi

Rjajtj

Rjajtj

Rjaj

Rjaj

Aaajaj

Rj

10 ,1 ,0 ,0

10 ,1 ,

110 ,1 ,

1 50

,

,

,

, s.t.

Max

11

1

n

n

n

23

Inferring an ELECTRE TRI model from assignment examples

p,...,,tk,...,iqk

p,...,,tk,...,iqp

p,...,,tk,...,ippbgbg

,.

Aayk

b,aCk

Aaxk

b,aCk

Aay

Aax

y,x

tii

ti

ti

ti

tititi

Rjajk

i i

ki tjii

Rjajk

i i

ki tjii

Rjaj

Rjaj

Aaajaj

Rj

10 ,1 ,0 ,0

10 ,1 ,

110 ,1 ,

1 50

,

,

,

, s.t.

Max

11

1

1

1

1 1

n

n

n

n

n

Nonlinear optimization problem with

2r+3n(p+1)+n+2 variables4r+3n(p+1)+2 constraints

24

Inferring an ELECTRE TRI model from assignment examples

Approximation of marginal concordance indices Ci(aj, bt)

Ci(aj, bt) are piecewise linear functions (non-differentiable)

We approximate Ci(aj, bt) by a differentiable sigmoidal function f(x): 01

1xxexp

xf

25

Inferring an ELECTRE TRI model from assignment examples

The approximation of Ci(aj, bt) by f(x) is such that x0=(pjt+qj

t)/2, and

(which influences the angle of inclination of Ci(aj, bt) in point (x0,0.5))

is chosen such that surface of overestimation A is equal to surface of

underestimation B – then the approximation error is minimal

= 5.55/ (pjt-qj

t)

^

qj-pit -qi

t(-pit-qi

t)/2

gi(ai)-pit

Ci(aj,bt)Ci(aj,bt)^

2555

1

1ti

ti

tijiti

ti

tjiqp

bgagqp

.exp

b,aC

26

Inferring an ELECTRE TRI model from assignment examples

Illustrative example:

There are 3 categories C1, C2, C3, separated by 2 limit profiles b1 and

b2, defined on 3 criteria g1, g2, g3 with an increasing scale [0, 100]

27

Inferring an ELECTRE TRI model from assignment examples

Optimization problem has 23 variables and 44 constraints

Additional constraints , prevent any criterion to be

a dictator

Determination of the starting point: ki = 1, i=1,…,3

where nt is the number of reference actions assigned to Ct

31 ,21 3

1,...,ikk

i ii

p,...,t,...,in

ag

n

ag

bgt

Caji

t

Caji

titjtj 1 ,31 ,

21

1

1

28

Inferring an ELECTRE TRI model from assignment examples

Initial values of indifference and preference thresholds:

qit = 0.05 gi(bt), i=1,…,n, t=0,1,…,p

pit = 0.1 gi(bt), i=1,…,n, t=0,1,…,p

= 0.75

qi2

pi2

qi1

pi1

29

Inferring an ELECTRE TRI model from assignment examples

Initial values of xaj and yaj:

Inconsistency: a4 is assigned to C1, while DM assigned a4 to C2

= xa4 = –0.083 < 0

30

Inferring an ELECTRE TRI model from assignment examples

Moreover, = 0.37, =0.629, k1=0.517, k2=1, k3=0.483

qi2

pi2

qi1

pi1

31

Inferring an ELECTRE TRI model from assignment examples

Final values of xaj and yaj:

Model M is consistent for all [0.629–0.37, 0.629+0.37],

thus for all [0.5, 1]

This proves the model is robust

32

ELECTRE TRI Assistant

ELECTRE TRI Assistant permits to infer:

weights ki, i=1,…,n

cutting level

Limit profiles, indifference and preference thresholds are fixed

Assuming , the regression problem is a Linear

Programming problem

ni ik1 1